A cross-intersection theorem for vector spaces based on semidefinite programming

TL;DR

This paper establishes a new upper bound on the product of sizes of families of subspaces with nontrivial intersections in a finite vector space, using semidefinite programming techniques inspired by Lovász's theta function.

Contribution

It introduces a novel semidefinite programming approach to derive a cross-intersection theorem for vector spaces, providing explicit solutions and characterizing extremal families.

Findings

Derived an upper bound for the product of family sizes with intersection properties.

Constructed explicit optimal solutions to the semidefinite program.

Characterized the structure of extremal families.

Abstract

Let $\mathscr{F}$ and $\mathscr{G}$ be families of $k$- and $\ell$-dimensional subspaces, respectively, of a given $n$-dimensional vector space over a finite field $\mathbb{F}_q$. Suppose that $x \cap y \ne 0$ for all $x \in \mathscr{F}$ and $y \in \mathscr{G}$. By explicitly constructing optimal feasible solutions to a semidefinite programming problem which is akin to Lov\'{a}sz's theta function, we show that $|\mathscr{F}| |\mathscr{G}| \leq {n-1 \brack k-1} {n-1 \brack \ell-1}$, provided that $n \geq 2k$ and $n \geq 2\ell$. The characterization of the extremal families is also established.